基于三角模糊数的软件项目风险评估.doc

译文西 安 邮 电 学 院毕 业 设 计（论 文）题 目： 基于三角模糊数的软件项目风险评估 院 （系）： 通信与信息工程学院 专 业： 电子信息科学与技术 班 级： 电科0703班 学生姓名： 范文超 导师姓名： 兰蓉 职称： 讲师 起止时间： 2011年1月3日至2011年6月10日 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 1, FEBRUARY 200345Fuzzy Risk Analysis Based on Similarity Measuresof Generalized Fuzzy NumbersShi-Jay Chen and Shyi-Ming Chen, Senior Member, IEEEAbstractIn this paper, we present a new method for fuzzyrisk analysis based on similarity measures of generalized fuzzy numbers. Firstly, we present a method called the simple center of gravity method (SCGM) to calculate the center-of-gravity (COG) points of generalized fuzzy numbers. Then, we use the SCGM to propose a new method to measure the degree of similarity between generalized fuzzy numbers. The proposed similarity measure uses the SCGM to calculate the COG points of trapezoidal or triangular generalized fuzzy numbers and then to calculate the degree of similarity between generalized fuzzy numbers. We also prove some properties of the proposed similarity measure and use an example to compare the proposed method with the existing similarity measures. The proposed similarity measure can overcome the drawbacks of the existing methods. We also apply the proposed similarity measure to develop a new method to deal with fuzzy risk analysis problems. The proposed fuzzy risk analysis method is more flexible and more intelligent than the existing methods due to the fact that it considers the degrees of confidence of decisionmakers opinions.Index TermsCenter-of-gravity (COG) points, fuzzy risk anal-ysis, generalized fuzzy numbers, similarity measures.I. INTRODUCTIONHE traditional center-of-gravity (COG) method 25 isIn this paper, we also present a new method to calculate thedegree of similarity between fuzzy numbers based on COG points of fuzzy numbers. The proposed similarity measure can overcome the drawbacks of the existing methods. Based on the proposed similarity measure, we present a new method for fuzzy risk analysis based on the similarity measure of general-ized fuzzy numbers. The proposed fuzzy risk analysis method is more flexible and more intelligent than the existing methods due to the fact that it considers the degrees of confidence of decisionmakers opinions.The rest of this paper is organized as follows. In Section II, we briefly review the definitions of generalized fuzzy numbers and their arithmetic operations 3, 4, the traditional COG) method 2, 14, 21 and the existing similarity measures of fuzzy numbers 7, 17, 26. In Section III, we present an SCGM to calculate the COG points of generalized fuzzy num-bers. In Section IV, we present a new method to calculate the degrees of similarity between generalized fuzzy numbers based on the COG points of generalized fuzzy numbers. Furthermore, we also prove some properties of the proposed similarity mea-sure. We also use an example to compare the proposed similarity measure with the existing methods. In Section V, we use the pro-Tvery useful to deal with the defuzzification problems 2,posed similarity measure of generalized fuzzy numbers to deal21, 31-33 and the fuzzy ranking problems 10, 11 byusing the COG points. However, there are some drawbacks in the traditional COG method, i.e., it cannot directly calculate the COG point of a crisp interval or a real number, and it is very time-consuming to calculate the COG point. In 8, we have presented a new method, called the simple center-of-gravity method (SCGM), to calculate the COG points of fuzzy numbers based on the concepts of plane vectors and linear equations. The proposed SCGM method can overcome the drawbacks of the traditional COG method.To measure the similarity of fuzzy numbers is very important in the research topic of fuzzy decision-making 6, 16, 26 and fuzzy risk analysis 7, 19, 29. Some methods have been presented to calculate the degree of similarity between fuzzy numbers 7, 17, 26. However, there are some drawbacks in the existing similarity measures, i.e., they cannot correctly calculate the degree of similarity between two generalized fuzzy numbers in some situations.Manuscript received February 10, 2002; revised May 28, 2002 and June 27, 2002. This work was supported in part by the National Science Council, Re-public of China, under Grant NSC 90-2213-E-011-054.The authors are with the Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, Republic of China (e-mail: .tw).Digital Object Identifier 10.1109/TFUZZ.2002.806316with the fuzzy risk analysis problems. The conclusions are dis-cussed in Section VI.II. PRELIMINARIESIn this section, we briefly review the concepts of generalized fuzzy numbers and their arithmetic operations 3, 4, the tradi-tional COG method 2, 14, 21, and some existing similarity measures of fuzzy numbers 7, 17, 26.A. Generalized Fuzzy Numbers and Their Arithmetic OperationsIn 3 and 4, Chen represented a generalized trapezoidal fuzzy number as , where , and, and are real numbers. The generalized fuzzy numberis a fuzzy subset of the real line , whose membership functionsatisfies the following conditions:1)is a continuous mapping from to the closed interval0,1;2), where;3)is strictly increasing on ;4), where;5)is strictly decreasing on ;6), where 1063-6706/03$17.00 2003 IEEE A and 46IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 1, FEBRUARY 2003Fig. 1. Two generalized trapezoidal fuzzy numbersB.If, then the generalized fuzzy numberis called anormal trapezoidal fuzzy number denoted. Ifand, then is called a crisp interval. If, thenis called a generalized triangular fuzzy number. Ifand, then is called a real number.Fig. 1 shows two different generalized trapezoidal fuzzy num-bers and which de-note two different decisionmakers opinions. The valuesandrepresent the degrees of confidence of the opinions of the decisionmakers and , respectively, where andBecause the traditional fuzzy arithmetic operations only can deal with normalized fuzzy numbers, they not only change the type of membership function of fuzzy number after arithmetical operations, but also have a drawback of requiring troublesome and tedious arithmetical operations. Thus, in 3, Chen pro-posed the function principle, which could be used as the fuzzy arithmetical operations between generalized fuzzy numbers, where these fuzzy arithmetical operations can deal with the generalized fuzzy numbers (i.e., nonnormal fuzzy numbers). In 18, Hsieh et al. pointed out that fuzzy arithmetical operators presented in 3 are not only do not change the type of mem-bership function of fuzzy number after arithmetical operations, but also can reduce the troublesomeness and tediousness of arithmetical operations. Thus, in this paper, we use Chens fuzzy arithmetical operators shown in (1)-(5) to deal with the fuzzy arithmetical operations between generalized fuzzy numbers. Assume that there are two generalized trapezoidal fuzzy numbers and whereand. The arithmetic operationsbetween the generalized trapezoidal fuzzy numbersandare reviewed from 3, 4, and 18 as follows.iii) Fuzzy Numbers Multiplication:(3)where, and. It is obvious that if,and are allpositive-real numbers, then(4)iv) Fuzzy Numbers Division:The inverse of the fuzzy numberis, where, andareall nonzero positive-real numbers or all nonzero negative-real numbers. If , , , , , , , and are all nonzeropositive real numbers, then the division ofandis(5)The difference between the generalized fuzzy numbers arith-metic operations and the traditional fuzzy numbers arithmetic operations is that the farmer can deal with both nonnormalized and normalized fuzzynumbers, but the latter only can deal with normalized fuzzy numbers.B. Traditional Center of Gravity MethodThe traditional COG method 2, 21 is very useful to deal with defuzzification problems and fuzzy ranking problems. The formula for calculating the center-of-gravity of a fuzzy number is shown as follows:(6)whereis the membership function of the fuzzy number,indicates the membership value of the element in,andIn 11, Cheng transforms (6) into another formula reviewed as follows. Assume that there is a trapezoidal fuzzy number ,whereand the membership functionof thetrapezoidal fuzzy number is shown as follows: i) Fuzzy Numbers Addition:otherwise(1)whereis continuous and strictly in-where, andare any real numbers.creasing;is continuous and strictly de-ii) Fuzzy Numbers Subtraction:creasing(7)(2)andwhere, andare any real numbers.(8)CHEN AND CHEN: FUZZY RISK ANALYSIS BASED ON SIMILARITY MEASURESThen, the transformed formula of (6) is shown as follows:(9)The traditional COG method can be used to deal with the de-fuzzification problems 2, 21, 31-33 and the fuzzy ranking problems 10, 11 by the COG point ( ). However, thereare some drawbacks in the traditional COG method. According to (9), we can see that it cannot directly calculate the COG of a crisp interval or a real number due to the fact that the de-nominators of (7) and (8) will become zero. Furthermore, it is very time consuming to calculate the COG point of a triangular or a trapezoidal fuzzy number. Thus, we must develop a new COG method to overcome the drawbacks of the traditional COG method.C. Similarity Measures Between Fuzzy NumbersAssume that there are two trapezoidal fuzzy numbers, whereand, then the degreeof similaritybetween the trapezoidal fuzzy numbersand can be calculated as follows 7:(10)47where is the universe of discourse(13)and(14)The larger the value of, the more the similarity betweenthe trapezoidal fuzzy numbers and. For example, assumethat, and, then based on (12), we can get the equation shown at the bottom of the page.In 17, Hsieh et al. proposed a similarity measure using the “graded mean integration-representation distance,” where the degree of similarity between fuzzy numbers andcan be calculated as follows:(15)whereandare thegraded mean integration representations ofand, respec-tively. Ifand are triangular fuzzy numbers, whereand, then the graded mean integra-tion representationsandof and, respectively,are defined as follows 17:where. Ifbers, wheredegree of similaritylated as follows 7:The larger the value of the fuzzy numbers andand are triangular fuzzy num-and, then thebetween and can be calcu-(11), the more the similarity between(16)(17)If and are trapezoidal fuzzy numbers, then the graded mean integration representations and of and , respec-tively, are defined as follows 17:(18)(19) In 26, Lee proposed a similarity measure for trapezoidalfuzzy numbers and used the similarity measure to deal withfuzzy opinions for group decision making, where the degree of similarity between the trapezoidal fuzzy numbersand,and, can becalculated as follows:(12)The larger the value of, the more the similarity betweenthe fuzzy numbers andIII. A NEW SIMPLE CENTER OF GRAVITY METHOD (SCGM) In this section, we present a new method, called SCGM, to calculate the COG point of a generalized fuzzy number. The proposed SCGM method is based on the concept of the medium curve 30. In the following, we briefly review the concept of 48Fig. 2. Function(x) of medium curve of the generalized trapezoidal fuzzynumber A.the medium curve from 30. Assume that there is a generalized trapezoidal fuzzy number , the definition ofthe medium curve is defined as follows 30.Definition 3.1: A medium curve of the fuzzy number is a function shown as follows:if(20)IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 1, FEBRUARY 2003Fig. 3. Gravity G of a triangle.Fig. 4. Gravity G of a triangle. otherwise.where;is called the-cut of the fuzzy numberand is defined as30;) denotesthe lower bound of21;) denotes the upper boundof21;denotes themiddle point of the-cutof the fuzzy numberFor example, Fig. 2 shows the generalized trapezoidal fuzzynumber with the functionof medium curve, where themedium curve is a straight line, and there are two points ()and () defined as follows:(21)(22)From Fig. 2, we can see that the formula of the medium curve is as follows:(23)where the linear equation shown in (23) is the medium curve of the generalized trapezoidal fuzzy numberIn the following, we briefly review the center of gravity of a triangle and the center of gravity of a rectangle, and we just use the symmetric example for easy to illustrate. Then, we propose the SCGM method to calculate the center of gravity points of generalized fuzzy numbers.a) COG of a Triangle: Fig. 3 shows a triangle. From Fig. 3, we can see that the center of gravity of thetriangle is on the medium curve denoted by a dotted line, where(24)(25)Becauseand, we can see that(26)whereFig. 5. Gravity T of a rectangle.Let us consider an asymmetrical triangle as shown in Fig. 4.From Fig. 4, we can see that the center of gravity of a symmetric triangle is ( ), where and are shown in(24) and (25), respectively.b) COG of a Rectangle: Fig. 5 shows a rectangle. From Fig. 5, we can see that the center of gravity of therectangle is on the medium curve, where(27)(28)whereIt is obvious that the center of gravity of a trapezoidal is be-tween the center of gravity of a triangle and the center of gravity of a rectangle as shown in Fig. 6.From Fig. 6, we can see that the valueof the trapezoidalCOG point is located between the valueof the triangularCOG point (i.e.,) and the valueof the rectangularCOG point (i.e.,). Thus, the range of the valueofa trapezoidal COG point is as follows:whereFrom Figs. 3, 5, and 6, when we assume that a triangle is a tri-angular fuzzy number, a rectangle is a crisp interval, and a trape-zoidal is a trapezoidal fuzzy number, we can see that the COG CHEN AND CHEN: FUZZY RISK ANALYSIS BASED ON SIMILARITY MEASURESFig. 6. Center of gravity of a trapezoid is between the gravity of a triangle and the gravity of a rectangle.point of a generalized fuzzy number is on the medium curve. If we use the value of a COG point and the medium curve,we can obtain the value of the COG point. Based on the pre-vious discussions, we can propose a new COG method called the SCGM described as follows. If is a generalized trapezoidal fuzzy number, where , then, we cansee that the valueof the COG point of is as follows:ifandifand(29)Ifis a generalized triangular fuzzy number, where, we can see that the valueof the COG point of is as follows:Ifis a crisp interval, where, then wecan see that the valueof the COG point of is as follows:Based on (21), (22), and (23), we can obtain the valueof theCOG point of as follows:(30)where,and. Thus, (30) can be rewritten as(31)Based on (29) and (31), we can obtain the COG pointof a generalized trapezoidal fuzzy number, where49Fig. 7. Three generalized triangular fuzzy numbers u , u , and u .In the following, we use an example presented in 10 and 11 to illustrate how to use the proposed SCGM method to cal-culate the COG point of a generalized fuzzy numbers. Assume that there are three generalized triangular fuzzy numbers , ,and as shown in Fig. 7.The generalized triangular fuzzy numbercan be repre-sented as. According to (29), wecan calculate the valueof the COG pointof thegeneralized triangular fuzzy number, where, as follows:Then, we can use the valueand (31) to calculate the valueof the COG pointof the generalized triangularfuzzy number shown as follows:Thus, the COG point of the generalized triangular fuzzy number is . In the sameway, we can obtain the COG pointsandof the generalized triangular fuzzy numbersand, respec-tively, shown as follows:, and. Thus, we can see thatand. If we use thetraditional COG method, we also can obtain the same results. However, the proposed method can overcome the drawbacks of the traditional COG method, i.e., the traditional COG method cannot directly calculate the center-of gravity of a crisp interval or a real number, and it is very time consuming to calculate the COG point. For example, assume that there are three different types of generalized fuzzy numbers , , and as shown inFig. 8, where the generalized fuzzy number is a crisp interval; the generalized fuzzy number is a trapezoidal fuzzy number; the generalized fuzzy number is a real number. 50IEEE TRANSACTIO